Problem-solving plays a fundamental role in mathematics and is central in any mathematician’s life. Since the very early days of civilisation, mathematical problem-solving has featured in the creation of counting, commerce and geometry, and in the more recent past, it has spurred the technological and economical advances made by society. But what does it entail?
Consider the following problem (adapted from Zeitz (2007) The Art and Craft of Problem Solving): a couple celebrating their anniversary invited four couples for a party. The host (let us name him Person A) asked all present how many people they shook hands with (this took place pre-COVID-19!). Everyone questioned shook hands with a different number of people, and of course, none shook hands with their respective partner. How many people did the hostess (the wife of Person A) shake hands with?
The problem is very easy to understand. Yet, the non-problem-solver might be tempted to turn the page when faced with such a problem, easily becoming uninterested in trying to find a solution maybe as a result of not even knowing from where to start.
On the other hand, an experienced problem-solver is confident that a (not necessarily unique) solution can be found, and, if not, why such a problem might not be solvable.
A solution to the above problem can be obtained by first listing all the possible different number of handshakes. The nine people questioned made the following number of handshakes: 0, 1, …, 8 (remember that no one shook hands with his/her partner). It might be useful to respectively name the people P0, P1, …, P8, according to their respective number of handshakes. So, P8 must be the partner of P0 (do you see why?).
Repeating the process (maybe a diagram would help) would eventually lead to the number of handshakes made by the hostess (four). The same argument can be repeated for any number n of couples. However, proving the general case involves what in mathematics is called an inductive argument, first used by Euclid (c. 300BC) to prove that there is an infinite number of primes.
Critical thinking is central and must be practised throughout the mathematical problem-solving process. It involves four main phases (see the illustration): the problem must be clearly understood, a realistic plan is devised, the plan is carried out systematically, and the whole process is re-examined and reflected upon, possibly restarted if necessary.
John Baptist Gauci is a senior lecturer at the Department of Mathematics within the Faculty of Science of the University of Malta.
Sound bites
• A group of computer scientists at Carnegie Mellon University developed a new tool that turns complex mathematical equations into visualisations. Different from a graphing calculator, this software named ‘Penrose’ creates illustrations of complex equations arising from various mathematical fields. It will be formally presented at the SIGGRAPH2020 Conference on Computer Graphics and Interactive Techniques next month. See also http://penrose.ink/.
• In summer 2019, Lisa Piccirillo answered in the negative a problem originally posed more than 50 years ago by John H. Conway on whether the Conway knot is a slice of a higher-dimensional knot. Slice knots are used to investigate the strange nature of four-dimensional space, in which two-dimensional spheres can often be knotted in ways that they cannot be smoothed out.
For more sound bites, listen to Radio Mocha: Mondays at 7pm on Radju Malta and Thursdays at 4pm on Radju Malta 2 (https://www.fb.com/RadioMochaMalta).
Did you know?
• Mathematical problem-solving skills took a considerable boost since the publication of George Pólya’s book How to Solve It in 1945. Pólya (1887-1985) was a Hungarian mathematician whose works in combinatorics, numerical analysis, number theory and probability theory remain highly influential till the present day. Translated into several languages, the book sold over a million copies.
• Various mathematical competitions are held worldwide, all having one theme in common: the ability to solve mathematical problems. The most prestigious is the annual International Mathematical Olympiad (IMO). The first IMO was held in Romania in 1959 and today it attracts the participation of students coming from over 100 counties. Participants are under 20 years and not registered at any tertiary institution. The content of the problems covers a vast array of topics and solving these questions requires a high degree of ingenuity and rigorous training.
For more trivia, visit www.um.edu.mt/think.